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Is it possible to find a lagrangian for a system with a varible mass and have a vaild solution when you are complete?

For instance, if I have a chain falling over the edge of a table or a rocket how would one approach this.

Example of my thinking:

For a chain falling over a table, if we assume that a sufficent portion of the entire mass is hanging over the edge (M'), than the lagrange will be something along the lines of:

L=1/2 (dm/dq)*v(t)^2 + M'gl; where q is a generalized coordinate, and l is how high the object is above the ground-level.

Anyone see anything wrong with this picture? Any other examples?

For instance, if I have a chain falling over the edge of a table or a rocket how would one approach this.

Example of my thinking:

For a chain falling over a table, if we assume that a sufficent portion of the entire mass is hanging over the edge (M'), than the lagrange will be something along the lines of:

L=1/2 (dm/dq)*v(t)^2 + M'gl; where q is a generalized coordinate, and l is how high the object is above the ground-level.

Anyone see anything wrong with this picture? Any other examples?

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